# AC70050E Analyse and critically evaluate risk return considerations in relation to the application of portfolio theory and management

Financial Risk Management

Our module will discuss the different classifications of risk and we will place emphasis on financial risk management. The importance to understand the concepts of declined profits, increased losses and negative returns is essential in risk evaluation processes.

Most of the treatment here is of "mathematical" flavour. The complicity of the models discussed is simplified by examining the applied applications of their usage rather than their theoretical implications. We discuss theoretical concepts in a much lighter format. Mathematical tools are essential for good understanding of the course`s contents, but nothing to worry about!

I shall prepare you for the mathematical contents of the syllabus. I will be covering matrix algebra, probability, probability distributions, financial theory (forward and option contracts, Markowitz portfolio theory) and the introduction to the Black - Scholes Formula together with its application. We also present the Binomial model.

We will also use the computer laboratory to conduct the second half of this syllabus. I will be concentrating my efforts at demonstrating the market applications of the learned concepts. We will discuss one and two asset returns and their volatilities (risk) in some detail.
Module aims:

• To provide students with an understanding of the underlying mathematics involved in assessing market risk.
• To provide students with an appreciation of practical risk assessment using Excel-based financial models.

Summary of Contents

The content of the module will include, but will not necessarily be limited to, nor in the provided order, the following topics:

• Introductory probability theory - Probability Distributions: the Binomial distribution, the Normal distribution.
• Matrix algebra: the use of matrices in Linear and non-linear systems and Optimisation Models.
• Portfolio Theory and Management: Risk, return, correlation, covariance in portfolios combining risky and risk-free assets, Markowitz and the Two - Asset portfolio theory.
• Options, Futures, and Other Derivatives: the Binomial Model, the Black-Scholes Formula
• Computer Applications and Applied Financial Analysis: Portfolio Theory & spreadsheet applications, Calculations of returns & volatility

• Financial risk analysis. VaR, Conditional VaR, historical simulation, stress testing.

Learning outcome 1: Demonstrate an understanding of the probability theory and matrix algebra in relation to the fundamentals of risk management.

Learning outcome 2: Analyse and critically evaluate risk return considerations in relation to the application of portfolio theory and management

Learning outcome 3: Demonstrate an understanding of the mathematical principles that underpin the assessment of financial market risk

Learning outcome 4: Develop and interpret spreadsheet models for financial market risk assessment.

Assessment 1

Probability

1. The discrete random variable X has the following probability distribution:

 x 1 2 3 4 5 P(X=x) 1 4 1 4 1 8 1 8 1 4

Find (a) P (X≤2) (b) P(X>2) (c) P(1

2. The discrete random variable Y has the following probability distribution:

 r 1 2 3 4 P(Y=r) 0.3 0.1 0.4 0.2

Find E(Y) and Var(Y)

3. The discrete random variable Z has the following probability distribution:

 r 1 2 3 4 5 P(Z=r) 2 10 0.35 10 100 3 20 K

Find the value of K.

4. A fair cubical die has a 1 on two sides, a 2 on three sides and a 3 on the other side. The random variable S represents the score obtained when the die is rolled.
(a) Write down the probability distribution of S.
(b) Find E(S) and Var(S).

5. A cubical die is biased such that the score is proportional to the number showing eg P(5) = 5k etc. Find the probability distribution for S, the score on the die.

6. The random variable T has the following distribution: (a,b are constants)

 r 1 2 3 P(T=r) a b a

(a) Work out E(T)

(b) Given Var(T) = 0.75, work out values of a and b. (5 marks)

Portfolio related

1. One share of stock costs £55 and has a beta of 0.75. Investors anticipate that the stock will pay a year-end dividend of £3. We are also informed that the current Treasury bill rate is 5%. while the market risk-premium stands at 9.7 per cent.

(a) Please establish what investors` expectations of the stock price at the end of the year will be assuming a fairly priced stock today.
(b) Now let us conjecture that investors` beliefs change and they now think the stock will sell for £57 at the end of the year.
1. Would you consider this stock good or bad investment?
2. What are investors likely to do? Explain
3. What in your view will the new equilibrium price be in which the stock will be perceived as fairly priced?

2. A financial analyst uses the following information. A 5 per cent rate from Treasury bills, a market portfolio`s expected return of 14 per cent. Based on the CAPM formulation she needs to establish:
(a) what the market risk premium will be
(b) with a beta of 1.5, what the required return on an investment will be
if subsequently the market expects a return of 12 per cent from Exige Inc.`s stock, calculate its beta coefficient

3. Nersess UK, a spinoff of a global retail brand, has invested £16 million in long-term corporate bonds. The expected annual rate of return for this bond portfolio is 6 per cent, and the annual standard deviation is 10 per cent.
One prominent financial advisor for the company, proposes that Nersess UK consider investing in an index fund, which closely tracks the FTSE 250 index. 14% and 16% are the expected return and standard deviation of the index respectively.
a. Now assume that Nersess UK puts all it`s money in a combination of the index fund and Treasury bills. Is it possible for its expected rate of return to improve without causing a change in the portfolio`s risk? The Treasury bill yield is a riskless 4.5%.
b. Assume that correlation between the index fund and the corporate bond portfolio is 0.2. Could you deduce that Nersess UK might improve their performance by assigning equal investment amounts between the index fund and the corporate bond portfolio?

4. Given an efficient portfolio, let its expected return be 19% and its standard deviation 5%. Also assume the risk-free rate to be 5% and the expected return on the market portfolio of risky assets 25%.

In such a market environment, please find what expected rate of return a security would make if it had a 0.6 correlation with the market and a standard deviation of 3%.

5. An efficient portfolio has the following parameters: rp = 22%, rf = 5%, rm = 15%, and σm = 18%.
a. Obtain its beta and its standard deviation
b. Find its correlation with the market

Assessment 2: Required: Production of a Report Assignment brief

The assignment will be a report based around portfolio management and Excel modelling.

Historical data on share prices for several FTSE100 companies has been collected and can be seen in an excel file that will be posted on Blackboard one month prior to the submission deadline. The data denote the share price at the beginning of each week over several years.
Firstly, you are required to carry out some statistical analysis on this data to measure the expected return and the level of risk for each share, and the correlations between the returns on the various shares.

Secondly you are required to complete the following 6 analyses. Most importantly, you need to explain your results and show an understanding of the underlying theory and limitations of your analysis.

Required analyses:

1. Working with the file of share prices, estimate the weekly and the annual expected return from each share.

2. Estimate the standard deviation of the annual returns.
(Note that annual variance = weekly variance x 52, but annual SD = weekly SD x √52)

3. Produce a matrix of correlations between the weekly returns on each pair of shares.

4. Assuming that the appropriate risk-free rate is currently 2% and using as inputs your previous answers for expected return, standard deviation, and correlation, create a spreadsheet model which will identify efficient portfolios of the shares, and from this derive the efficient frontier chart. Explain, in the context of share portfolios the relationship between risk and return, and how diversification reduces risk.

5. Discuss the practical limitations of this approach to portfolio management and of the model itself.

6. Based upon the volatility of the tangency portfolio, calculate the 95% VaR of this portfolio. Comment upon how this measure may not be a good estimate of the VaR as time progresses while the portfolio remains the same. 